いえ generally refers to the physical house, while うち refers to the idea of "home", although the two being reversed isn't uncommon (just like in English).

うち also means "inside" (内) and can refer to someone's inner social circle (adding to the "home" meaning). Therefore, it can feel more intimate than いえ. Also, うち is very commonly used to mean the most central of that circle: "me" (usually used by women).

Rich was right with the general usage of うち only being you're own home, but there are still many exceptions. My girlfriend often says "おうち" to refer to the home of the person she is talking to.

Kitsune: As with all Japanese sentences, if it is understood from context then you don't need to specify who's home. If you see only うち in an "invitation" sentence, it's safe to assume that is the home of the speaker because they wouldn't be inviting the other person to go to the other person's house. If it is too ambiguous, just using うち would not be sufficient enough and you would need watashi no.